I want to learn about fractals. What should I read first?
"Chaos" is a good book to get a general overview and history. "Fractals Everywhere" is a textbook on fractals that describes what fractals are and how to generate them, but it requires knowing intermediate analysis. "Chaos, Fractals, and Dynamics" is also a good start. There is a longer book list at the end of this file (see "What are some general references?").
Also, use networked resources.
What is a fractal? What are some examples of fractals?
A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale.
There are many mathematical structures that are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. Fractals also describe many real-world objects, such as clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes.
Benoit Mandelbrot gives a mathematical definition of a fractal as a set for which the Hausdorff Besicovich dimension strictly exceeds the topological dimension. However, he is not satisfied with this definition as it excludes sets one would consider fractals.
According to Mandelbrot, who invented the word: "I coined 'fractal' from the Latin adjective 'fractus'. The corresponding Latin verb 'frangere' means 'to break' to create irregular fragents. It is therefore sensible - and how appropriate for our needs! - that, in addition to 'fragmented' (as in 'fraction' or 'refraction'), 'fractus' should also mean 'irregular', both meanings being preserved in 'fragment'."
-- "The Fractal Geometry of Nature", page 4
What is chaos?
Chaos is apparently unpredictable behavior arising in a deterministic system because of great sensitivity to initial conditions. Chaos arises in a dynamical system if two arbitrarily close starting points diverge exponentially, so that their future behavior is eventually unpredictable.
Weather is considered chaotic since arbitrarily small variations in initial conditions can result in radically different weather later. This may limit the possibilities of long-term weather forecasting. (The canonical example is the possibility of a butterfly's sneeze affecting the weather enough to cause a hurricane weeks later.)
Devaney defines a function as chaotic if it has sensitive dependence on initial conditions, it is topologically transitive, and periodic points are dense. In other words, it is unpredictable, indecomposable, and yet contains regularity.
Allgood and Yorke define chaos as a trajectory that is exponentially unstable and neither periodic or asymptotically periodic. That is, it oscillates irregularly without settling down.
What is fractal dimension? How is it calculated?
A common type of fractal dimension is the Hausdorff-Besicovich Dimension, but there are several different ways of computing fractal dimension.
Roughly, fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. The differences come in what is exactly meant by "object size" and what is meant by "measurement scale" and how to get an average number out of many different parts of a geometrical object. Fractal dimensions quantify the static *geometry* of an object.
For example, consider a straight line. Now blow up the line by a factor of two. The line is now twice as long as before. Log 2 / Log 2 = 1, corresponding to dimension 1. Consider a square. Now blow up the square by a factor of two. The square is now 4 times as large as before (i.e. 4 original squares can be placed on the original square). Log 4 / log 2 = 2, corresponding to dimension 2 for the square. Consider a snowflake curve formed by repeatedly replacing ___ with _/\_, where each of the 4 new lines is 1/3 the length of the old line. Blowing up the snowflake curve by a factor of 3 results in a snowflake curve 4 times as large (one of the old snowflake curves can be placed on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal.
What is topological dimension?
Topological dimension is the "normal" idea of dimension; a point has topological dimension 0, a line has topological dimension 1, a surface has topological dimension 2, etc.
For a rigorous definition:
A set has topological dimension 0 if every point has arbitrarily small neighborhoods whose boundaries do not intersect the set.
A set S has topological dimension k if each point in S has arbitrarily small neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the least nonnegative integer for which this holds.
What is a strange attractor?
A strange attractor is the limit set of a chaotic trajectory. A strange
attractor is an attractor that is topologically distinct from a periodic orbit
or a limit cycle. A strange attractor can be considered a fractal attractor.
An example of a strange attractor is the Henon attractor.
Consider a volume in phase space defined by all the initial conditions a
system may have. For a dissipative system, this volume will shrink as the
system evolves in time (Liouville's Theorem). If the system is sensitive to
initial conditions, the trajectories of the points defining initial
conditions will move apart in some directions, closer in others, but
there will be a net shrinkage in volume. Ultimately, all points will
lie along a fine line of zero volume. This is the strange attractor. All
initial points in phase space which ultimately land on the attractor
form a Basin of Attraction. A strange attractor results if a system is
sensitive to initial conditions and is not conservative.
Note: While all chaotic attractors are strange, not all strange attractors
are chaotic.
What is the Mandelbrot set?
The Mandelbrot set is the set of all complex c such that iterating z -> z^2+c does not go to infinity (starting with z=0).
Click here for an image of the Mandelbrot set.
How is the Mandelbrot set actually computed?
Why do you start with z=0?
What are the bounds of the Mandelbrot set? When does it diverge?
How can I speed up Mandelbrot set generation?
What is the area of the Mandelbrot set?
What can you say about the structure of the Mandelbrot set?
Is the Mandelbrot set connected?
What is the difference between the Mandelbrot set and a Julia set?
What is the connection between the Mandelbrot set and Julia sets?
How is a Julia set actually computed?
What are some Julia set facts?
How does complex arithmetic work?
How does quaternion arithmetic work?
What is the logistic equation?
What is Feigenbaum's constant?
What is an iterated function system (IFS)?
What is the state of fractal compression?
How can you make a chaotic oscillator?
What are laboratory demonstrations of chaos?
What are L-systems?
What is some information on fractal music?
How are fractal mountains generated?
What are plasma clouds?
Where are the popular periodically-forced Lyapunov fractals described?
What are Lyapunov exponents?
How can Lyapunov exponents be calculated?
How can 3-D fractals be generated?
What is Fractint?
How does Fractint achieve its speed?
Where can I obtain software packages to generate fractals?
Where are fractal pictures archived?
Where can I obtain fractal papers?
What is complexity?
What are some general references on fractals and chaos?
What are some relevant journals?
Are there any special notices?
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